1. Calibration factor
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1 Annex_C_MUBDandP_eng_.doc, p. of pages Annex C: Measureen uncerainy of he oal heigh of profile of a deph-seing sandard ih he sandard deviaion of he groove deph as opography er In his exaple, he uncerainy sources derived in general in chaper of Annex A are ransferred o he condiions of deph-seing sandards. In he exaple a glass sandard ih a noinal value of µ is assued.. Calibraion facor. Reference sandard According o he calibraion cerificae, he uncerainy of he reference sandard calibraed dephseing sandard is, for exaple, U n = n, a k =, U u P = n n = n 4. Difference easuring race calibraion race a G y u P y =. Wih a y = µ and G = n/ Annex A Figure, μ n / u P y = =, n. Repeaabiliy As esiaed value for he uncerainy in he racing of he reference sandard, he experienal sandard deviaion of he ean value of P is aken fro repea easureens perfored a he sae poin: u s P b = = s P. Wih a ypical value for he experienal variance s P = n is u b = 4 n.. Topografy of he sandard In he direcion of he longiudinal axis, he groove deph of he es objec o be easured is usually no consan. A ypical value is a variaion idh of 4 n. This is hy he groove deph is deerined on differen poins of he groove. As esiaed value for he uncerainy coponen of he profile poins, he sandard deviaion of he ean value fro P is used: s P s P =. A prerequisie of his observaion is ha he groove deph varies randoly and no syseaically. In he case of ypically = profile seps, P has an experienal sandard deviaion of n. This eans ha in he case of his observaion, he less labour-inensive disseinaion of he ean
2 Annex_C_MUBDandP_eng_.doc, p. of pages value of he groove deph is of ain ineres, and an exacer posiioning of he easuring poin is dispensed ih. s P u z = e = n. Sraighness deviaion of he reference For he guidance devaiions in he guiding area used for he device, W n is assued. The variance conribuion is n u zref = = n. = W 4. Background noise The background noise of an opical fla be Rz = n. u zpl = n. = Rz. Plasic deforaion For a glass sandard, he plasic deforaion is negligibly sall. 6. Uncerainy of he poins of he oal profile Fro he su of hese uncerainies, he variance of he poins of he oal profile is obained as follos: U a u n y z g = 4 s P s P W Rz The firs hree conain he uncerainy of he raceabily and he las hree he uncerainy conribuions fro he easuring process. The suands of his equaion are lised in he able in chaper 9. Wih he exeplary nuerical values fro chapers o 4 he folloing is obained: u z g =. n.
3 Annex_C_MUBDandP_eng_.doc, p. of pages 7. Paraeer funcion 7. Uncerainy of easureen of he oal heigh of profile P In he aligned profile, he oal heigh of profile P is he difference beeen he highes z-value on he reference plane z h and he loes z-value in he read of he groove z l. When he alignen is ade, he profile poins are lifed differenly and an alignen deviaion occurs due o roughness and flaness errors. By his, he difference beeen he highes and he loes poin varies by A a os, as a funcion of here hese lie. Taking he average of several easureens, he expecaion value is zero. The odel for P is valid: P = z h z l A. A A A: reference leveling A: reference leveling A: leveling deviaion P : leveling P: leveling Pr : roughness on reference plane Figure 7: Uncerainy of P due o alignen deviaion The folloing is valid for he uncerainy of he oal heigh of profile P: P P P u P = u zh u zl u A. z z A h l The o firs sensiiviy coefficiens are. The hird describes in principle he cosinue facor hich changes ih he alignen. Due o he saller angles and heir changes, i is equal o one ih sufficien accuracy. The uncerainy of he z-values uz h and uz l is equal o ha of he poins of he overall profile uz g.. The peak value P r of he roughness or he flaness deviaion of he profile coponens in he reference line secions included in he evaluaion serves as an esiaed value for he variabiliy of he alignen deviaion ua. Wihin P r, a recangular disribuion is assued. u =. P u z g Pr
4 Annex_C_MUBDandP_eng_.doc, p. 4 of pages On deph-seing sandards, P r = n is a usual value. The variance of P hus is deerined as u P =, n n, and he sandard uncerainy of P is u P = 4, 6 n Wih he coverage facor k=, an expanded uncerainy of easureen of U P = 9. n is obained. 7. Measureen uncerainy of he groove deph D A A A D Pr A: reference line ih levelling error A: assued correc reference line A: levelling error Pr : roughness on reference plane D : deph of groove Figure 8: Uncerainy of D due o alignen deviaion As an approxiaion for he calculaion of he groove deph D in accordance iho DIN EN ISO 46-, he folloing odel is assued. On he aligned profile, he difference of he ean values of he "op profile secions" and he "boo profile secion" are fored Annex A Figure. If he reference line secions are aligned ih he alignen deviaion A, he profile ill be disored and D ill be changed. The odel: For he ean of various easureens, he expeced value of A is zero. Figure 8: Uncerainy of D by alignen deviaion nh n l D = zghi z n n h i = l i = gli A. For he uncerainy, he su rule is valid nh n D l D D u D = u zghi u zgli u A. z z A i = ghi i = gli
5 Annex_C_MUBDandP_eng_.doc, p. of pages D = z ghi n h for i = o n h, D = z gli n l for i= o n l, u z, u z = u z. ghi gli g For he gvien sall angles and heir changes, he cosine facor accuracy. D is equal o ih sufficien A In conras o he analysis of he uncerainy of P, he folloing observaion is valid as esiaed value for ua: Due o he definiion of D sapling in he cenre and on he edge of he profile secion, he affec of he alignen deviaion only aouns o half he conribuion of P r. Pr D = u zg nh nl u. Averaging over he profile poins only acs on he rando deviaions in z g so ha he sandard uncerainy u D = U a n y 4 G s D s P W P Rz r [ n h n l ud = 8. n. Wih he coverage facor k =, an expanded uncerainy of easureen of UD = 7 n is obained. ] 8. Evaluaion ih filering For he evaluaion ih λs, he uncerainy of he poins of he priary profile is varied by he soohing facor of he filer funcion f s as a funcion of he he shor-ave lo-pass filer λs and he spacing of he easuring poins Δx Annex A, chaper 4. λs /µ Δx /µ f s Table : Filer facors for differen lo-pass filer avelenghs. The filer facor f s only affecs he currenly easured rando quaniies so ha he folloing is valid for he uncerainy of he poins of he priary profile:
6 Annex_C_MUBDandP_eng_.doc, p. 6 of pages U a u n y z s = 4 W [ f s s P s P Rz ] If he sae values as in chapers o 4 are used for he inpu quaniies, he folloing is obained for filering ih λs = 8 µ and Δx =. µ: uz s = 8. n. In he pracical calculaion, he naure of he inpu quaniies is o be aken ino accoun: If he esiaed values for he inpu quaniies P, P r, W, Rz se fro profile daa already filered, hey include already he effec of he shor-ave lo-pass filer. If he filered values are aken for he calculaion of uz s in accordance ih he above equaion, he filer facor f s hus us no be applied again. The calculaion schee hen becoes he sae as in chaper 6, only ih he values of he filered inpu quaniies. 8. Uncerainy of he oal heigh of profile P Fro he uncerainy of he profile poins and he alignen deviaion u P = u z s Pr = a { U n y 4 W s f s P s P Rz } P r and up =.7 n. 8. Uncerainy of he groove deph D As in chaper 7., he algorih of D averages secion by secion over he filered profile poins. u D = u z s P r n n h l The averaging only affecs he rando deviaions in z s so ha U a u n y D = 4 W Pr f s [ s P s D Rz n h n l ] and ud = 8. n 9. Suary of he easureen uncerainy of oal heigh of profile P and groove deph D For he exeplary values, a deph-seing sandard ih he noinal value P = µ as assued and for he inpu quaniies, ypical values ere used.
7 Annex_C_MUBDandP_eng_.doc, p. 7 of pages Toal heigh of profile P, evaluaion ihou filering Caper Inpu quaniy cachord. Reference sandard. Difference easuring poin calibraion poin. Repeaabiliy Calculaion of inpu quaniy Exeplary value Sensiiviy coeff. U U n = n B n 4 a y a y = µ G=n/ s P Mehod of deerinaion, disribuion B Recangular s P = n = Variance /n². Topography over groove lengh s P s P = n = Guidance deviaion 4 Background noise 6 Variance of he profile poins 6 Uncerainy of he profile poins 7. Uncerainy of oal heigh of profile Expanded uncerainy of oal heigh of profile P W Rz Su of his colun [ u P r z g ] W = n B Recangular Rz = n Recangular u. z g Uncerainy /n uz g. P r = n, u P 4.6 k up UP 9.
8 Annex_C_MUBDandP_eng_.doc, p. 8 of pages Groove deph D, evaluaion ihou filering Caper Inpu quaniy cachord. Reference sandard. Difference easuring poin calibraion poin. Repeaabiliy Calculaion of inpu quaniy Exeplary value Sensiiviy coeff. U U n = n B n 4 a y a y = µ G=n/ s D s D = n = Mehod of deerinaion, disribuion B Recangular Variance /n². Topography over groove lengh s P s P = n = Guidance deviaion 4 Background noise W n h n l Rz 7. Profile alignen P r 7. Variance of he groove deph 7. Uncerainy of he groove deph Expanded uncerainy of groove deph D Su of his colun W = n B Recangular Rz = n n h, n l = Recangular P r = n Recangular.7. u D 7. Uncerainy /n ud 8. k ud UD 7
9 Annex_C_MUBDandP_eng_.doc, p. 9 of pages Toal heigh of profile P, evaluaion ih filering λs = 8 µ Chaper Inpu quaniy cachord. Reference sandard. Difference easuring poin calibraion poin. Repeaabiliy Topography over groove lengh Guidance deviaion 4 Background noise 8 Variance of he profile poins 8 Uncerainy of he profile poins 8. Uncerainy of oal heigh of profile P Expanded uncerainy of oal heigh of profile P Calculaion of inpu quaniy Exeplary value Sensiiviy coeff. U U n = n B n 4 a y a y = µ G=n/ s P f s s P f s W f s Rz Su of his colun [ u P r z g ] Mehod of deerinaion, disribuion B Recangular s P = n = s P = n = W = n B Recangular Rz = n Recangular Variance /n²... u z s 68. Uncerainy /n uz s 8. P r = n, u P.7 k up UP.4
10 Annex_C_MUBDandP_eng_.doc, p. of pages Groove deph D, evaluaion ih filering λs = 8 µ Chaper Inpu quaniy cachord. Reference sandard. Difference easuring poin calibraion poin. Repeaabiliy Calculaion of inpu quaniy Exeplary value Sensiiviy coeff. U U n = n B n 4 a y a y = µ G=n/ f s s D s D = n = Mehod of deerinaion, disribuion B Recangular Variance /n².. Topograph over groove lengh Guidance deviaion 4 Background noise f s s W fs Rz P n h n l 8. Profile alignen P r 8. Variance of he groove deph 8. Uncerainy of he groove deph Expanded Uncerainy of groove deph D Su of his colun s P = n = W = n B Recangular Rz = n n h, n l = Recangular P r = n Recangular.6. u D 6.9 Uncerainy /n ud 8. k ud UD 6.
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